What is a number state?

A number state in quantum mechanics is a quantum state that is represented by a vector in the Hilbert space of a quantum system, where the vector is a function of a continuous variable that represents a number, such as the position or momentum of a particle. The probability of finding the system in a particular number state is given by the square of the magnitude of the vector.

Number states are often used to describe the quantum states of particles in a harmonic oscillator potential. In this case, the number states are labeled by a quantum number $n$, which represents the number of quanta of energy that the particle has. The higher the value of $n$, the more energy the particle has.

Number states can also be used to describe the quantum states of particles in other types of potentials. For example, they can be used to describe the quantum states of electrons in atoms. In this case, the number states are labeled by a set of quantum numbers that include the principal quantum number $n$, the angular momentum quantum number $l$, and the magnetic quantum number $m$.

Number states are an important concept in quantum mechanics because they provide a way to represent the quantum states of particles in a way that is both mathematically rigorous and physically intuitive.